用MMD优化量化，通过梯度流实现更鲁棒的粒子分布逼近。
Weighted quantization using MMD: From mean field to mean shift via gradient flows
- 基于MMD设计粒子系统的梯度流，支持可变权重
- 提出MSIP算法，比现有方法在高维多峰数据上更稳定
- 统一了均值漂移、梯度下降与聚类算法，适合复杂分布建模
使用一组粒子近似概率分布是机器学习与统计中的基础问题，应用于聚类和量化。形式上，目标是寻找一组带权重的Dirac测度混合，以最佳逼近目标分布。尽管现有工作多依赖Wasserstein距离衡量误差，最大均值差异（MMD）却较少被关注，尤其在允许粒子权重变化时。我们主张，Wasserstein-Fisher-Rao梯度流非常适合于设计基于MMD最优的量化方案。我们证明，满足一组常微分方程的相互作用粒子系统可离散化该梯度流。进一步提出一种新型不动点算法——均值漂移交互粒子（MSIP）。我们表明，MSIP扩展了经典的均值漂移算法，广泛用于核密度估计中模式识别。此外，MSIP可解释为预条件梯度下降，并作为Lloyd聚类算法的一种松弛。该统一框架将梯度流、均值漂移与MMD最优量化结合，经高维与多模态数值实验验证，其鲁棒性优于当前最先进方法。
原文摘要 · Abstract (English)
Approximating a probability distribution using a set of particles is a fundamental problem in machine learning and statistics, with applications including clustering and quantization. Formally, we seek a weighted mixture of Dirac measures that best approximates the target distribution. While much existing work relies on the Wasserstein distance to quantify approximation errors, maximum mean discrepancy (MMD) has received comparatively less attention, especially when allowing for variable particle weights. We argue that a Wasserstein-Fisher-Rao gradient flow is well-suited for designing quantizations optimal under MMD. We show that a system of interacting particles satisfying a set of ODEs discretizes this flow. We further derive a new fixed-point algorithm called mean shift interacting particles (MSIP). We show that MSIP extends the classical mean shift algorithm, widely used for identifying modes in kernel density estimators. Moreover, we show that MSIP can be interpreted as preconditioned gradient descent and that it acts as a relaxation of Lloyd's algorithm for clustering. Our unification of gradient flows, mean shift, and MMD-optimal quantization yields algorithms that are more robust than state-of-the-art methods, as demonstrated via high-dimensional and multi-modal numerical experiments.
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